A Convexity Theorem for Local mean values of Subtemperatures
A Convexity Theorem for Local mean values of Subtemperatures
复制标题
局部低温平均值的凸性定理
DOI：
10.1112/blms/22.3.245
复制
发表时间：
1990
影响因子：
0.9
通讯作者：
N. Watson
中科院分区：
文献类型：
--
作者：
N. Watson
次谐波函数的特点是其在球体上的积分均值不等式，毫无疑问球体是最好使用的表面。然而，热方程的子解已使用两个不同表面上的方法以类似的方式进行表征。通常使用矩形的边界，如[3，p.1]中所示。 277]，它们的优点是矩形的泊松核是已知的，至少是一系列的和。然而，在[11]中，使用了基本解的水平面（下面称为基本面），其动机来自于 Pini [9] 和 Fulks [6] 提出的温度均值定理。对于大多数目的来说，这两种方法同样有效。但在建立热方程的维纳准则时，基本表面的使用是必不可少的[5]。在本文中，我们研究了基本表面上的低温均值的性质。矩形表面的相应陈述要么是未知的，要么是明显错误的。定理 1 是温度均值的线性定理，类似于半径为 r 的球体上的环面上的调和函数的均值形成 log r 或 r2"" 的线性函数的经典结果。定理2是对应的低温均值凸性定理。在定理 3 中，我们针对某些体积均值得出了类似的凸性结果，并建立了体积均值和表面均值之间的不等式。作为定理2的应用，我们给出了Pini-Fulks定理的推广，并证明了基本面边界区域上低温热延续的唯一性。
Subharmonic functions are characterized by an inequality involving their integral means over spheres, and there is no question that spheres are the best surfaces to use. Subsolutions of the heat equation, however, have been characterized in an analogous manner using means over two different surfaces. Usually the boundaries of rectangles are used, as in [3, p. 277], and these have the advantage that a Poisson kernel for rectangles is known, at least as the sum of a series. However, in [11], level surfaces of the fundamental solution (called fundamental surfaces below) were used, the motivation coming from a mean value theorem for temperatures due to Pini [9] and Fulks [6]. For most purposes the two approaches are equally good; but in establishing a Wiener criterion for the heat equation, the use of fundamenial surfaces is essential [5].In this paper we study the properties of the means of subtemperatures over fundamental surfaces. The corresponding statements for rectangular surfaces are all either unknown or demonstrably false. Theorem 1 is a linearity theorem for the means of temperatures, analogous to the classical result that the means of harmonic functions on an annulus over spheres of radius r form linear functions of log r or r2"". Theorem 2 is a corresponding convexity theorem for the means of subtemperatures. In Theorem 3, we derive a similar convexity result for certain volume means, and establish inequalities between the volume and surface means. As applications of Theorem 2, we give an extension of the Pini-Fulks theorem, and prove the uniqueness of thermic continuation of subtemperatures over domains bounded by fundamental surfaces.